Story highlights

New PM will need to keep the economy from diving

Leader will have to bring squabbling party member together

London (CNN)In case you haven't noticed, Britain is in a bit of a pickle. The British people voted last week to pull out of the European Union, and as a result, they have no clue who's going to lead the country through one of its toughest political transitions in years. And most will have no say in it, either. Oops.

British Prime Minister David Cameron, who campaigned to remain in the 28-country union, resigned the day after the "Brexit" vote, saying he would step down in October.

How will Britain choose its next prime minister?

Britain has a parliamentary system, so unlike in the United States, people in Britain do not directly elect their national leader. They elect members of Parliament to represent them, and then the party, or coalition of parties, that holds the majority of seats in Parliament take government. Their leader then becomes prime minister.

So when a prime minister resigns, he or she must be replaced. And that's a decision made among the party MPs and party members.

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Members of Cameron's Conservative Party are seeking the prime ministership. They need the backing of at least two MPs to officially run.

When there are three or more candidates, MPs vote in an initial round, and keep holding rounds until the number is whittled down to two. A final vote goes out to the wider party, and the winner is Britain's new prime minister.

What's on the new PM's to-do list?

A lot. The new prime minister will be ushering the country through one its most serious political storms in decades. On the top of the list will be the painstaking task of striking a deal on the terms of its exit from the EU, which is still reeling from last week's slap-in-the-face referendum.

- Strike a trade deal

So far, this is a mess. Cameron and EU leaders are still arguing over when to officially begin the process. Cameron says he wants to know what a deal will look like first, and the EU says talks can't begin until they are given confirmation that Britain wants out.

Britain will want access to Europe's tariff-free single market, but to do so, it is likely to be forced to accept free movement of labor -- the rule that EU citizens can turn up in Britain to look for work without a job offer, and vice versa. This could be a major problem, as stopping EU citizens from coming into the country in large numbers was a key argument in the campaign to leave the union.

It will be no easy task to get a good trade deal and manage an immigration policy that will keep the prime minister popular with British Brexiteers.

- Keep the economy afloat

A top priority will be making sure Britain's economy doesn't go under, or fall into recession.

British markets and the pound took a battering following the Brexit vote, but they do appear to be bouncing back. Of greater worry is the country's debt, now over £1,000,000,000,000 -- yep, that's a trillion pounds ($1.3 trillion), or 90% of GDP. Following the Brexit vote, the country lost its top credit ratings with agencies Standard & Poor's and Fitch Group, meaning investors are losing confidence that Britain can manage its debts.

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The prime minister will have to devise a plan to stop companies from fleeing the country, particularly from London, where many have set up their global or European headquarters. Telecom giant Vodafone and budget airliner easyJet, for example, have said they're thinking about finding a new home.

European cities are already advertising that they are open for business as alternative hubs. Niamh Bushnell, Dublin's commissioner for start-ups, saw Brexit as an opportunity for the Irish tech hub.

"Thanks to Brexit we have a new opportunity to attract Europe's (and Russia's and ...) serial or first time entrepreneurs to set up shop in Dublin," she said in a statement.

- Avoid a 'Screxit'

While Britain's Brexit has caused concerns of tearing the EU apart, there are worries that it could also cause a Screxit -- a Scottish exit -- from the UK. The Scots are among Britain's most fervent EU supporters, with 62% of voters supporting Britain remaining in the bloc. Leaders there have clearly said that another independence referendum is on the table, as they feel they had a clear mandate from their people to stay in the union. Scotland voted to remain part of the UK in 2014, with 55% of people voting against independence. Nonetheless, the movement has steam, and the new British prime minister may be tempted to give even more power the Scots, who already enjoy a large amount of autonomy in governance.

- Fight terrorism

On top of all this, Europe is facing its biggest terror threat in a generation. While security has largely remained in the domain of individual states, EU nations may reconsider the sharing of information with Britain after a withdrawal. The new prime minister will have to make sure this is not putting the country at greater risk of a terror attack. The current threat level is already at "severe."

- Stop the squabbling

And perhaps the most trying task of the new prime minster will be holding the Conservatives together. Infighting is what led to Cameron calling the referendum in the first place, a move that spectacularly backfired on the leader. If the party was split over whether to stay or leave the union, it will likely be split on many of the key issues, like trade and migration.

Who has thrown their hat in the ring?

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The favorite to replace Cameron was the eccentric former London Mayor Boris Johnson, one of the most prominent campaigners to Brexit. But he delivered a bombshell on Thursday, announcing he wouldn't run.

Here's who's left:

- Theresa May

The 59-year-old home secretary is the bookies' favorite to take over for Cameron. She has been behind strong but controversial security and counter-terrorism policies, and oversees the country's border control, a key issue in the Brexit debate. May backed remaining in the European Union but is still Euroskeptic in many ways, and has committed to exiting the union, despite voices growing louder to stall the process. May opened her campaign promising strong leadership and presenting a clear list of priorities.

- Michael Gove

Gove was a controversial education secretary, making radical reforms that earned him as many enemies as friends. He was eventually demoted by his good friend Cameron to chief whip, a position that requires keeping MPs in line with party policy and making sure they turn up to vote.

The 48-year-old politician's relationship with Cameron's turned sour during the referendum campaign period, as Gove stood on the opposing side, backing a Brexit.

Gove was born and brought up in Scotland, and worked for a while as a newspaper columnist, making frequent radio and television appearances. A man who attracts frequent attention for being unfashionable, Gove recently sent the Twittersphere into a frenzy by performing a Wham! rap to young school children.

- Stephen Crabb

Who? A lot of British people have no clue who Crabb is, even though he oversees the country's pensions policy. But he is reportedly well-liked among fellow Conservative MPs and has sought to distinguish himself form other Conservative politicians who are seen as out of touch with the people because they attended elite schools and come from upper-class backgrounds.

Crabb, 43, describes his roots as working-class, having grown up in Wales to a single mother in a social housing complex. He previously served as Welsh secretary, and has said he would make immigration control a priority, should he be given the chance to negotiate a deal with EU leaders.

- Andrea Leadsom

Leadsom, the country's energy and climate change minister, campaigned strongly for leaving the European Union, arguing that Britain had a strong enough economy to thrive without belonging to the EU.

The 53-year-old, with a background in banking and fund management, campaigned against the free movement of labor, calling for greater border control, a key issue in the Brexit debate. She also claimed that wages would rise if Britain stopped the flow of EU citizens turning up in the country looking for work.

- Liam Fox

Fox has had a stab at party leadership before. He finished third in the 2005 race, in which he lost out to Cameron before the Conservatives took power from center-left Labour.

The 54-year-old former defense secretary campaigned for Britain to leave the EU and has warned against backpedaling, as the separation process has failed to really kick off and as doubts are raised over whether it will actually happen. He campaigned strongly against the free movement of labor, vowing to implement stronger border control if appointed prime minister.

Fox resigned as defense secretary in 2011 in a lobbying scandal involving a close friend.

First day on the job

To get the ball rolling on Britain's EU divorce, the country must trigger Article 50 of the Lisbon Treaty, the pact which binds it to the union.

Cameron has passed the buck on this one, saying it is up to the new prime minister to do it. And the new leader will have quite a first day on the job -- the EU has said it wants the article invoked within 24 hours of the prime minister's appointment. But Britain won't necessarily budge. It is so far acting in its own good time.